The length of diagonal of a square is 9√2 cm, the square is reshaped to form a triangle. What is the area (in cm²) of largest incircle that can be formed in that triangle?

Solution (By Examveda Team)

a√2 = 9√2
a = 9

a + b + c = 36
Circle of maximum radius is only exist in equilateral Δ
3a = 36
a = 12
$$\eqalign{ & S = \frac{{a + b + c}}{2}, \cr & r = \frac{{\frac{{\sqrt 3 }}{4} \times {a^2}}}{{\frac{{12 + 12 + 12}}{2}}} \cr & \Rightarrow r = \frac{{\sqrt 3 }}{4} \times \frac{{12 \times 12 \times 2}}{{36}} \cr & \Rightarrow r = 2\sqrt 3 \cr} $$
∴ Area of circle = πr² = 12π